In this work we present some results for the inverse problem of the
identification of a single rigid body immersed in a fluid governed by the
stationary Boussinesq equations. First, we establish a uniqueness result.
Then, we show the way the observation depends on perturbations of the rigid
body and we deduce some consequences. Finally, we present a new method for
the partial identification of the body assuming that it can be deformed only
through fields that, in some sense, are finite dimensional. In the proofs, we
use various techniques, related to Carleman estimates, differentiation with
respect to domains, data assimilation and controllability of PDEs.